A357943 a(0) = 0; a(1) = 1, a(2) = 2; for n > 2, a(n) is the number of times the term a(n - 1 - a(n-1)) has appeared in the sequence.

0, 1, 2, 1, 1, 3, 1, 1, 5, 5, 5, 1, 3, 3, 3, 6, 3, 5, 5, 5, 5, 1, 7, 1, 1, 9, 5, 9, 8, 8, 9, 9, 1, 4, 2, 10, 4, 10, 4, 2, 2, 3, 3, 4, 4, 4, 7, 4, 7, 7, 7, 7, 7, 7, 8, 8, 7, 9, 9, 9, 9, 9, 9, 9, 4, 11, 4, 11, 9, 12, 12, 12, 12, 12, 12, 12, 12, 9, 13, 2, 13, 2, 6, 8, 8, 8, 13, 8, 6, 3, 3, 8, 9, 9

Offset: 0

Author

Scott R. Shannon, Oct 22 2022

Keywords

• nonn
• look

Comments

In the first 100 million terms the longest run of consecutive equal terms is eight : a(69)..a(76) = 12. There is no other run of equal length in this range, and it is unknown if a longer run exists.

Other than the initial three terms, the first time a term exists that is one more than the previous term is a(29) = 8, a(30) = 9. Remarkably the first time two such consecutive terms exists is not until a(60917874) = 5394, a(60917875) = 5395, a(60917876) = 5396. It is unknown if three or more such terms exist.

Note that if the sequence starts with just a(0) = 0, a(1) = 1 then the resulting sequence is A003056.

The sequence is conjectured to contain all positive numbers. See A357944 for the index of where a given number first appears.

Examples

			a(5) = 3 as the term at a(4 - a(4)) = a(4 - 1) = a(3) = 1, and 1 has appeared three times in the sequence.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 0..10000
• Scott R. Shannon, Image of the first 10000000 terms.
• Scott R. Shannon, Frequency distribution of the first 10000000 terms. The maximum x value is 4858, which is the number of times 20 has appeared.

Crossrefs

Cf. A357944, A003056, A357930, A342585, A181391.

Programs

Mathematica

nn = 83; c[] = 0; Array[Set[{a[#], c[#]}, {#, 1}] &, 3, 0]; Do[(a[n] = c[#]; c[c[#]]++) &[a[n - a[n - 1] - 1]], {n, 3, nn}]; Array[a, nn, 0] (* _Michael De Vlieger, Oct 23 2022 *)